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Properties

Label 2-7800-1.1-c1-0-30
Degree 22
Conductor 78007800
Sign 11
Analytic cond. 62.283362.2833
Root an. cond. 7.891977.89197
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.71·7-s + 9-s + 3.31·11-s + 13-s + 1.55·17-s − 5.33·19-s − 3.71·21-s + 0.442·23-s + 27-s + 2.56·29-s + 0.613·31-s + 3.31·33-s − 0.257·37-s + 39-s − 10.6·41-s + 12.6·43-s + 7.44·47-s + 6.78·49-s + 1.55·51-s − 5.39·53-s − 5.33·57-s − 13.1·59-s − 5.27·61-s − 3.71·63-s + 10.5·67-s + 0.442·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.40·7-s + 0.333·9-s + 1.00·11-s + 0.277·13-s + 0.377·17-s − 1.22·19-s − 0.810·21-s + 0.0923·23-s + 0.192·27-s + 0.477·29-s + 0.110·31-s + 0.577·33-s − 0.0423·37-s + 0.160·39-s − 1.65·41-s + 1.93·43-s + 1.08·47-s + 0.968·49-s + 0.218·51-s − 0.741·53-s − 0.706·57-s − 1.71·59-s − 0.675·61-s − 0.467·63-s + 1.28·67-s + 0.0533·69-s + ⋯

Functional equation

Λ(s)=(7800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 78007800    =    23⋅3⋅52⋅132^{3} \cdot 3 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 62.283362.2833
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7800, ( :1/2), 1)(2,\ 7800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1153923792.115392379
L(12)L(\frac12) ≈\approx 2.1153923792.115392379
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−T 1 - T
5 1 1
13 1−T 1 - T
good7 1+3.71T+7T2 1 + 3.71T + 7T^{2}
11 1−3.31T+11T2 1 - 3.31T + 11T^{2}
17 1−1.55T+17T2 1 - 1.55T + 17T^{2}
19 1+5.33T+19T2 1 + 5.33T + 19T^{2}
23 1−0.442T+23T2 1 - 0.442T + 23T^{2}
29 1−2.56T+29T2 1 - 2.56T + 29T^{2}
31 1−0.613T+31T2 1 - 0.613T + 31T^{2}
37 1+0.257T+37T2 1 + 0.257T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1−12.6T+43T2 1 - 12.6T + 43T^{2}
47 1−7.44T+47T2 1 - 7.44T + 47T^{2}
53 1+5.39T+53T2 1 + 5.39T + 53T^{2}
59 1+13.1T+59T2 1 + 13.1T + 59T^{2}
61 1+5.27T+61T2 1 + 5.27T + 61T^{2}
67 1−10.5T+67T2 1 - 10.5T + 67T^{2}
71 1−0.311T+71T2 1 - 0.311T + 71T^{2}
73 1−9.46T+73T2 1 - 9.46T + 73T^{2}
79 1−16.1T+79T2 1 - 16.1T + 79T^{2}
83 1−11.7T+83T2 1 - 11.7T + 83T^{2}
89 1+4.58T+89T2 1 + 4.58T + 89T^{2}
97 1−10.8T+97T2 1 - 10.8T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.898601872183510745471942063650, −7.07887586582490824219196118760, −6.37412077137627625529700486489, −6.15005817220053809787088867415, −4.95515973118666087053754639419, −4.03998480593778221413283022259, −3.54836061815997226057730920375, −2.79790544488079651684554345416, −1.87353633001064472785425594687, −0.69762297965155179851538211371, 0.69762297965155179851538211371, 1.87353633001064472785425594687, 2.79790544488079651684554345416, 3.54836061815997226057730920375, 4.03998480593778221413283022259, 4.95515973118666087053754639419, 6.15005817220053809787088867415, 6.37412077137627625529700486489, 7.07887586582490824219196118760, 7.898601872183510745471942063650

Graph of the ZZ-function along the critical line